MILAN (Reuters) - Telecom Italia held out a promise of richer shareholder returns hours after activist fund Elliott Advisors threatened to try to shake up the way top shareholder Vivendi runs the company.

Elliott said on Tuesday it had built an unspecified stake in TIM and was ready to replace some members of the board to improve strategy, value and governance at Italy's biggest phone group.

Setting out its plan for 2018-20, TIM sought to convince investors it could reverse years of sluggish share price performance and cope with new rivals that are appearing in both broadband and mobile.

TIM, which had net debt of more than 25 billion euros (22.40 billion pounds)at the end of 2017, last paid a dividend on ordinary shares five years ago, but new Chief Executive Amos Genish said the group may do so again if it meets the targets set out in the "DigiTIM" strategy.

Genish said TIM plans to make more of its processes digital, boost broadband coverage and offer more video, music and gaming content to win customers and grow its businesses.

Its board also approved a plan to put its network into a separate company fully controlled by TIM, in a move Genish hopes will help the market better value its assets.

"We have a winning strategy, a solid and ambitious plan," Genish said, adding he would be happy to sit down with Elliott and hear their input once they had a chance to study the plan.

Elliott is known for waging campaigns at companies which it sees as undervalued.

"When they go in, you know there's value to be extracted," said Banca Ifigest fund manager Roberto Lottici who bought TIM shares on the news.

Analysts said the new plan was ambitious, while the news around Elliott and the planned network separation increased the speculative appeal of the stock.

At 1224 GMT TIM shares were up 1.3 percent after rising around 6 percent on Tuesday.

YEARS OF TURMOIL

TIM has been going through a tumultuous three years since French media group Vivendi become its top investor and tightened its grip on the company. It eventually appointed two-thirds of TIM's board and named its own CEO as TIM's executive chairman.

Two chief executives left TIM after clashing with Vivendi and the French influence also led to confrontations with the Italian government in Rome.

TIM has lost more than one-third of its market value since Vivendi first took a stake in 2015.

The appointment of Genish, a favourite of Vivendi Chairman Vincent Bollore, left some investors guessing as to the French group's ultimate aim for TIM -- use it as a pillar of its plan to create a southern European media empire or as an asset to trade in the next wave of mergers in European telecoms.

In his plan Genish made no reference to any asset sales, saying TIM's Brazilian unit and Italian subsidiaries INWIT and Sparkle would support the company's growth to 2020.

The group will invest 9 billion euros in Italy over the next three years, and around 12 billion reais ($3.7 billion) in Brazil, mainly to boost ultrafast broadband coverage.

TIM expects to generate a total of around 4.5 billion euros in equity free cash flow over the three years.

TIM said domestic service revenues would remain broadly stable over the life of the plan, while core earnings would grow at a low single digit rate on average between 2017-2020.

For 2017, TIM said group revenues rose 4.2 percent to 19.8 billion euros, in line with expectations, while core earnings fell (EBITDA) 2.6 percent to 7.79 billion euros, mainly due to charges related to staff restructuring.

(Editing by Keith Weir)

By Agnieszka Flak and Stephen Jewkes

Stocks treated in this article : Vivendi, Telecom Italia, TIM Participacoes SA